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Markdown+itex2MML Sandbox

Play around below. Your changes will, periodically, be rolled back.

Some examples

(1)

min w_{h} p_{h} + w_{r} p_{r} + w_{l} p_{l}

Here’s an equation

(2)

\int_{- ∞}^{∞} e^{- x^{2} / 2} d x = \sqrt{2 π}

which we can later refer¹ back to as (2).

The Dirac equation:

(i ／ D + m) ψ = 0

Here’s the table of Clifford² algebras over $ℝ$ :

$j$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$𝒞 ℓ_{j}^{-}$	$ℝ$	$ℂ$	$ℍ$	$ℍ \oplus ℍ$	$ℍ (2)$	$ℂ (4)$	$ℝ (8)$	$ℝ (8) \oplus ℝ (8)$	$ℝ (16)$
$𝒞 ℓ_{j}^{+}$	$ℝ$	$ℝ \oplus ℝ$	$ℝ (2)$	$ℂ (2)$	$ℍ (2)$	$ℍ (2) \oplus ℍ (2)$	$ℍ (4)$	$ℂ (8)$	$ℝ (16)$

where the generators of $𝒞 ℓ_{j}^{\pm}$ satisfy

γ_{i} γ_{j} + γ_{j} γ_{i} = \pm 2 δ_{i j}

and $𝒞 ℓ_{n + 8}^{\pm} = 𝒞 ℓ_{n}^{\pm} \otimes ℝ (16)$ .

(3)

\lim_{n \to ∞} \sum_{k = 1}^{n} \frac{1}{k^{2}} = \frac{π^{2}}{6}

More Examples

(4)

\nabla \times \overset{⇀}{E} = - \frac{\partial \overset{⇀}{B}}{\partial t}

(5)

\oint B \cdot d l = μ_{0} I_{enc}

$H^{1} (𝒵, 𝒪 (- k))$ Let $G = (V, E)$ be a graph, with $w : V \to [0,1]$ a weight function.

(6)

{Q_{i}, Q_{j}} = δ_{ij} ℋ .

Here is a groupoid $𝒢$ . $3 (mod 5)$ .

Theorems

Definition

Let $H$ be a subgroup of a group $G$ . A left coset of $H$ in $G$ is a subset of $G$ that is of the form $x H$ , where $x \in G$ and $x H = {x h : h \in H}$ .

Similarly a right coset of $H$ in $G$ is a subset of $G$ that is of the form $H x$ , where $H x = {h x : h \in H}$ .

Lemma

Let $H$ be a subgroup of a group $G$ , and let $x$ and $y$ be elements of $G$ . Suppose that $x H \cap y H$ is non-empty. Then $x H = y H$ .

Proof

Let $z$ be some element of $x H \cap y H$ . Then $z = x a$ for some $a \in H$ , and $z = y b$ for some $b \in H$ . If $h$ is any element of $H$ then $a h \in H$ and $a^{- 1} h \in H$ , since $H$ is a subgroup of $G$ . But $z h = x (a h)$ and $xh = z (a^{- 1} h)$ for all $h \in H$ . Therefore $z H \subset x H$ and $x H \subset z H$ , and thus $x H = z H$ . Similarly $y H = z H$ , and thus $x H = y H$ , as required.

Lemma

Let $H$ be a finite subgroup of a group $G$ . Then each left coset of $H$ in $G$ has the same number of elements as $H$ .

Proof

Let $H = {h_{1}, h_{2}, \dots, h_{m}}$ , where $h_{1}, h_{2}, \dots, h_{m}$ are distinct, and let $x$ be an element of $G$ . Then the left coset $x H$ consists of the elements $x h_{j}$ for $j = 1,2, \dots, m$ . Suppose that $j$ and $k$ are integers between $1$ and $m$ for which $x h_{j} = x h_{k}$ . Then $h_{j} = x^{- 1} (x h_{j}) = x^{- 1} (x h_{k}) = h_{k}$ , and thus $j = k$ , since $h_{1}, h_{2}, \dots, h_{m}$ are distinct. It follows that the elements $x h_{1}, x h_{2}, \dots, x h_{m}$ are distinct. We conclude that the subgroup $H$ and the left coset $x H$ both have $m$ elements, as required.

Theorem

(Lagrange’s Theorem). Let $G$ be a finite group, and let $H$ be a subgroup of $G$ . Then the order of $H$ divides the order of $G$ .

Proof

Each element $x$ of $G$ belongs to at least one left coset of $H$ in $G$ (namely the coset $x H$ ), and no element can belong to two distinct left cosets of $H$ in $G$ (see Lemma 1). Therefore every element of $G$ belongs to exactly one left coset of $H$ . Moreover each left coset of $H$ contains $∣ H ∣$ elements (Lemma 2). Therefore $∣ G ∣ = n ∣ H ∣$ , where $n$ is the number of left cosets of $H$ in $G$ . The result follows.

Corollary

Let $x$ be an element of a finite group $G$ . Then the order of $x$ divides the order of $G$ .

Problems?

Maybe the notation should be different than the Latex counterparts, but these do not seem to be rendering correctly (JD: They look fine to me. A font problem in your browser?) (JB: That was exactly the problem, thanks for the suggestion. The braces weren’t stretching correctly with my old fonts.):

SVG:

K^{0} {\bar{K}}^{0}

Mixing

Complicated commutative diagrams (equations in SVG)

SVG in equations.

In $SU (3)$ , $\otimes = \oplus$ .

Cases:

r_{a + 1} = {\begin{array}{ll} 0 & with prob. \exp (- θ r_{a}) \\ \max {δ r_{a}, z} & with prob. 1 - \exp (- θ r_{a}) \end{array}

Stretchy Brackets:

q_{a} (z) = σ_{a}^{- 1} \exp [- \frac{γ + z}{σ_{a}}]

Linearity of Quadrature Rules

\sum_{i = 1}^{N} (α f (x_{i}) + β g (x_{i})) w_{i} = α \sum_{i = 1}^{N} f (x_{i}) w_{i} + β \sum_{i = 1}^{N} g (x_{i}) w_{i}

Linearity of Integrals

\int_{a}^{b} (α f (x) + β g (x)) dx = α \int_{a}^{b} f (x) dx + β \int_{a}^{b} g (x) dx

Can we talk about $x_{i}^{2}$ inline? What about $\int_{a}^{b} x^{2} dx$ ? Inline fractions $\frac{x - x_{2}}{x_{1} - x_{2}}$ ?
Big fractions

p_{3} (x) = (\frac{1}{2}) \frac{(x - \frac{1}{2}) (x - \frac{3}{4}) (x - 1)}{(\frac{1}{4} - \frac{1}{2}) (\frac{1}{4} - \frac{3}{4}) (\frac{1}{4} - 1)} + (\frac{1}{2}) \frac{(x - \frac{1}{2}) (x - \frac{3}{4}) (x - 1)}{(\frac{1}{4} - \frac{1}{2}) (\frac{1}{4} - \frac{3}{4}) (\frac{1}{4} - 1)} + (\frac{1}{2}) \frac{(x - \frac{1}{2}) (x - \frac{3}{4}) (x - 1)}{(\frac{1}{4} - \frac{1}{2}) (\frac{1}{4} - \frac{3}{4}) (\frac{1}{4} - 1)} + (\frac{1}{2}) \frac{(x - \frac{1}{2}) (x - \frac{3}{4}) (x - 1)}{(\frac{1}{4} - \frac{1}{2}) (\frac{1}{4} - \frac{3}{4}) (\frac{1}{4} - 1)}

Diagram

\begin{matrix} P_{1} (Y) & \to & P_{1} (X) \\ ↓ & ⇓^{\sim} & ↓ \\ T' & \to & T \end{matrix}

Will indented code work

This should be code
And this

yep

(7)

\begin{matrix}  \end{matrix} \equiv ({U (k)}^{n_{1}}, {v_{i}})

You can also refer to it as (2). Chacun à son goût!.
↩
For more information, see Wikipedia.
↩

Revised on November 21, 2009 22:34:11 by Jacques Distler (66.68.104.188)

Instiki Sandbox

Table of Contents (Skip the Table of contents)

Instiki

Markdown+itex2MML Sandbox

Some examples

More Examples

Theorems

Definition

Lemma

Proof

Lemma

Proof

Theorem

Proof

Corollary

Problems?

Instiki
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